Finite Group Databasebeta

← #20289994

Subgroups — #20289994

Order 6784.

270 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811593C211yesmaximal, characteristic
2C2×C22⋊#88324C211yesmaximal, characteristic
3#16811469C212yesmaximal, complemented
4#16811468C212yesmaximal, complemented
5C2×#13133194C211yesmaximal, characteristic
6C2×#11719940—21no—
7C22×#88324—21no—
8C22⋊#88324—21no—
9C22⋊#88324—21no—
10C22⋊#88324C2212yes—
11C22⋊#88324C2212yes—
12C22×Dic106C411yescharacteristic
13C23×Dic53C2211yescharacteristic
14#13133194—21no—
15#13133194—21no—
16#13133194—21no—
17#13133194—21no—
18C2×C4⋊Dic53C2211yescharacteristic
19C2×#11719919C411yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×C22⋊Dic53C411yescharacteristic
22C2×C22⋊C4×C53C411yescharacteristic
23C24×C53C2×C411yescharacteristic
24C22×C4×C53C2×C411yescharacteristic
25C22×C4×C53C2×C411yescharacteristic
26C22×Dic53—42no—
27C22×Dic53C2×C411yescharacteristic
28C22×Dic53C2×C411yescharacteristic
29C22×Dic53C2×C411yescharacteristic
30C22×Dic53D411yescharacteristic
31C22×Dic53D411yescharacteristic
32C22×Dic53D411yescharacteristic
33C22×Dic53D411yescharacteristic
34C22×Dic53C2311yescharacteristic
35C2×Dic106—21no—
36C2×Dic106—21no—
37C2×Dic106—42no—
38#11719940—42no—
39C2×#88324—22no—
40C2×#88324—22no—
41C2×#88324—42no—
42C22⋊C4×C53—21no—
43C22⋊C4×C53—21no—
44C22⋊Dic53—21no—
45C22⋊Dic53—21no—
46C4⋊Dic53—21no—
47C4⋊Dic53—21no—
48#11719919—21no—
49#11719919—21no—
50#11719919—21no—
51#11719919—21no—
52C23×C53—22no—
53C23×C53—22no—
54C23×C53—41no—
55C23×C53C22⋊C411yescharacteristic
56C23×C53C22⋊C411yescharacteristic
57C23×C53C22×C411yescharacteristic
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—22no—
61C2×C4×C53—41no—
62C2×Dic53—22no—
63C2×Dic53—22no—
64C2×Dic53—22no—
65C2×Dic53—21no—
66C2×Dic53—21no—
67C2×Dic53—22no—
68C2×Dic53—41no—
69C2×Dic53—41no—
70C2×Dic53—41no—
71C2×Dic53—41no—
72C2×Dic53C22⋊C411yescharacteristic
73C2×Dic53C22⋊C411yescharacteristic
74C2×Dic53C2×D411yescharacteristic
75C2×Dic53C2×D411yescharacteristic
76Dic106—42no—
77Dic106—42no—
78#88324—42no—
79#88324—42no—
80C22×C53—21no—
81C22×C53—21no—
82C22×C53—21no—
83C22×C53—21no—
84C22×C53—21no—
85C22×C53—21no—
86C22×C53—21no—
87C22×C53—44no—
88C22×C53#10112yes—
89C22×C53C2×C22⋊C411yescharacteristic
90C22×C53#13112yes—
91C4×C53—41no—
92C4×C53—41no—
93Dic53—21no—
94Dic53—21no—
95Dic53—21no—
96Dic53—41no—
97Dic53—41no—
98Dic53#9912yes—
99#1472—531nomaximal, Sylow
100C2×C53—22no—
101C2×C53—22no—
102C2×C53—41no—
103C2×C53#40811yescharacteristic
104C2×C53C2×#9911yescharacteristic
105C2×C53C2×#10111yescharacteristic
106C2×C22⋊Q8—531no—
107#407—531no—
108C2×C22⋊C8—531no—
109#324—532no—
110#324—532no—
111C53#147211yescharacteristic, complemented, Sylow
112C22×Q8—531no—
113C23×C4—531no—
114C2×M16—1061no—
115C22×C8—1061no—
116C22⋊Q8—1061no—
117C22⋊Q8—1061no—
118C22⋊Q8—1061no—
119C22⋊Q8—1061no—
120C2×C4⋊C4—531no—
121C2×C4⋊C4—531no—
122C2×C4⋊C4—531no—
123C2×C22⋊C4—531no—
124C2×C22⋊C4#615711yescharacteristic
125C22⋊C8—532no—
126C22⋊C8—532no—
127C22⋊C8—1061no—
128C22⋊C8—1061no—
129C24C2×#615711yescharacteristic
130C2×Q8—1061no—
131C2×Q8—1061no—
132C2×Q8—2122no—
133C22×C4—531no—
134C22×C4—531no—
135C22×C4—531no—
136C22×C4—531no—
137C22×C4—531no—
138C22×C4—531no—
139C22×C4—531no—
140C22×C4—531no—
141C22×C4—2122no—
142C22×C4C2×#615711yescharacteristic
143C22×C4C2×#615711yescharacteristic
144M16—2122no—
145C2×C8—1062no—
146C2×C8—1062no—
147C2×C8—2122no—
148C4⋊C4—1061no—
149C4⋊C4—1061no—
150C4⋊C4—1061no—
151C4⋊C4—1061no—
152C4⋊C4—1061no—
153C4⋊C4—1061no—
154C22⋊C4—21no—
155C22⋊C4—21no—
156C22⋊C4—1061no—
157C22⋊C4—1061no—
158C23—22no—
159C23—22no—
160C23—41no—
161C23C22⋊#615711yescharacteristic
162C23C22⋊#615711yescharacteristic
163C23C22×#615711yescharacteristic
164Q8—2122no—
165Q8—2122no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—531no—
171C2×C4—531no—
172C2×C4—531no—
173C2×C4—531no—
174C2×C4—1062no—
175C2×C4—1062no—
176C2×C4—1062no—
177C2×C4—1061no—
178C2×C4—1061no—
179C2×C4—1062no—
180C2×C4—2121no—
181C2×C4—2121no—
182C2×C4—2121no—
183C2×C4—2121no—
184C8—2122no—
185C8—2122no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—44no—
194C22#1313327812yes—
195C22C2×C22⋊#615711yescharacteristic
196C22#1313330412yes—
197C4—41no—
198C4—41no—
199C4—532no—
200C4—1061no—
201C4—1061no—
202C4—1061no—
203C4—2121no—
204C4—2121no—
205C2—22no—
206C2—22no—
207C2—41no—
208C2#1681159611yescharacteristic
209C2D53⋊#9911yescharacteristic
210C2C2×#1313327811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC23×C53C22×C4
FrattiniC23C22×#6157
FittingC2×C22⋊C4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC2×C22⋊C4#6157
hypocenterC53#1472
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C2×C22⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #16811469C2semidirect
4. #16811468C2semidirect
111. C53#1472semidirect